Simplify the following expression and state the condition under which the simplification is valid. You can assume that $y \neq 0$. $q = \dfrac{6(4y + 5)}{4} \div \dfrac{8y(4y + 5)}{5y} $
Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{6(4y + 5)}{4} \times \dfrac{5y}{8y(4y + 5)} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ 6(4y + 5) \times 5y } { 4 \times 8y(4y + 5) } $ $ q = \dfrac{30y(4y + 5)}{32y(4y + 5)} $ We can cancel the $4y + 5$ so long as $4y + 5 \neq 0$ Therefore $y \neq -\dfrac{5}{4}$ $q = \dfrac{30y \cancel{(4y + 5})}{32y \cancel{(4y + 5)}} = \dfrac{30y}{32y} = \dfrac{15}{16} $